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Physical Review A

Effective-potential expansion method for the many-body problem at finite temperatures. I. Basic formalism

Yasutami Takada, Takafumi Kita

Published Sep 1, 1990• 4 citations
Publisher

Abstract

The effective-potential expansion method, namely, a variational approach to the many-body problem at zero temperature, is extended to nonzero temperatures by the introduction of a new trial density matrix in the Gibbs variational principle for the grand potential \ensuremath{\Omega}. The trial density matrix contains an effective potential as a variational parameter. We have given an expansion form for \ensuremath{\Omega} in terms of the effective potential. Special care is taken to evaluate the entropy term correctly.

DOI
10.1103/physreva.42.3242

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The effective-potential expansion method, namely, a variational approach to the many-body problem at zero temperature, is extended to nonzero temperatures by the introduction of a new trial density matrix in the Gibbs variational principle for the grand potential \ensuremath{\Omega}. The trial density matrix contains an effective potential as a variational parameter. We have given an expansion form for \ensuremath{\Omega} in terms of the effective potential. Special care is taken to evaluate the entropy term correctly.
Following previous work in which a variational method for the many-body problem was proposed wherein an effective potential $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ was introduced as a variational parameter to construct a trial wave function, we apply the method to the two-dimensional electron gas. In the method, every physical quantity is expanded in powers of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ instead of the bare potential as in the usual perturbation-theoretic approach. Because of this choice of expansion parameter, the expansion series converges rapidly even in the strong-coupling region. The result for the correlation energy in the two-dimensional electron gas agrees very well with that given by the variational Monte Carlo method even in the lowest-order calculation (i.e., to second order in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$). The difference is within several percent for 1<rs<1001<{r}_{s}<1001<rs​<100.
A new variational approach is proposed for the study of the many-body problem with the purpose of combining the advantages of perturbation-theoretic and variational approaches, while avoiding their disadvantages. A trial wave function is constructed so that a general expansion for the expectation value of an arbitrary physical quantity can be derived. Each term in the expansion can be calculated with the use of the techniques of Feynman diagrams, but in contrast with usual perturbation-theoretic approaches, every term in the expansion always converges. The procedure for applying the method to real systems is illustrated by a calculation of the ground-state energy of the electron gas at high densities.
The variational density-matrix approach introduced by Campbell, K\"urten, Ristig, and Senger [Phys. Rev. B 30, 3728 (1984)], to study finite-temperature Bose fluids is reformulated to avoid dealing directly with the entropy of the trial density matrix. We obtain exact expressions for the Euler-Lagrange equations for a finite-temperature trial statistical density matrix, which is the simplest finite-temperature generalization of a Jastrow trial ground-state wave function. A multicomponent hypernetted-chain method is developed to solve approximately these Euler-Lagrange equations at low temperatures. In leading order the results are shown to be equivalent to those obtained by Campbell et al. The current formalism provides a scheme to improve results of the previous approach.
A one-dimensional many-electron system with a repulsive \ensuremath{\delta}-function interaction is studied by the application of the variational method developed in the preceding paper [Takada and Kita, Phys. Rev. A 42, 3242 (1990)] in order to illustrate its actual implementation. Our results on the grand potential, the entropy, and the specific heat are compared in detail with the exact ones that are calculated by the numerical solution of the coupled integral equations obtained by the Bethe ansatz.
A self-contained pedagogical introduction to the functional Schrödinger picture method of many-body theory is given at a level suitable for graduate students and also for many-body physicists who have not been exposed to the functional Schrödinger picture method previously. Mathematical tools necessary for the functional Schrödinger picture calculation are introduced. The method is first applied to various condensed matter problems including the electron gas, the Hubbard model, the BCS superconductivity, and dilute bose gas within the variational approximation. It is shown that the variational approximation with the Gaussian trial functional invariably leads into the Hartree-Fock results for both zero and finite temperature cases. In order to go beyond the Gaussian results, concepts of variational and optimized perturbation theories are introduced using simple quantum mechanical languages. Then, the variational perturbation theory in the functional Schrödinger picture is applied to the λϕ4λϕ^4λϕ4 model to yield the effective potential to the third order. The optimized perturbation theory is also applied to the λϕ4λϕ^4λϕ4 model to yield a general expression up to the second order.
A variational principle is presented for the ground state energy in terms of the one-particle density matrix. The functional expression of the energy in terms of the density matrix is obtained as the limit of zero temperature of the free energy. This is achieved by adding a nonlocal external potential to the N-body Hamiltonian and obtaining a renormalized equation for the density matrix instead of the Green's function. By eliminating the external source it was possible to express the energy as a functional of the density matrix.